To answer the OPs second question, the main cause of non-linearity in valves is physics. It happens to be the case that an ideal triode has a current which varies as the 3/2 power of the effective voltage (set by grid and anode voltage combined). A real triode will have additional non-linearity from things like finite grid wire size and winding pitch, grid rods distorting the fields, anode shape etc. Secondary emission is not an issue in triodes, but it is for tetrodes and pentodes.
A 3/2 power is not linear - that would require a power of 1. That means that input signals get multiplied together, as well as being simply amplified. The multiplication produces distortion products, as explained by others using trigonometry. When they teach you trig at school they don't tell you that this is essential to understanding audio distortion (the teacher might not know this!).
Fortunately, the 3/2 power law distortion can be avoided by using an infinite anode load, as the anode distortion then exactly cancels the grid distortion. A real finite load does this imperfectly, and in any case does not deal with the other non-linearities. This means that some degree of valve non-linearity is unavoidable.
A 3/2 power is not linear - that would require a power of 1. That means that input signals get multiplied together, as well as being simply amplified. The multiplication produces distortion products, as explained by others using trigonometry. When they teach you trig at school they don't tell you that this is essential to understanding audio distortion (the teacher might not know this!).
Fortunately, the 3/2 power law distortion can be avoided by using an infinite anode load, as the anode distortion then exactly cancels the grid distortion. A real finite load does this imperfectly, and in any case does not deal with the other non-linearities. This means that some degree of valve non-linearity is unavoidable.
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harmonics are produced by the tubes (or other active elements in the circuit) -
A single cycle of ac signal impresses a varying voltage on the grid- let's say between 0V and -10V
As the instantaneous value of the input signal moves between 0V and -1V, it may produce a 20V change in plate voltage. As the same signal moves between -7 and -8 volts, it may produce only a 15V change in plate voltage. So the output voltage is not exactly like the input, i.e. it's not just the input multiplied by some number. The "number' (gain) actually changes between the more positive regions of the input signal and the more negative regions. The amplifier is then said to be non-linear. This non-linearity is distortion, since the output waveform has a fundamentally different shape than the input waveform.
Make sense?
A single cycle of ac signal impresses a varying voltage on the grid- let's say between 0V and -10V
As the instantaneous value of the input signal moves between 0V and -1V, it may produce a 20V change in plate voltage. As the same signal moves between -7 and -8 volts, it may produce only a 15V change in plate voltage. So the output voltage is not exactly like the input, i.e. it's not just the input multiplied by some number. The "number' (gain) actually changes between the more positive regions of the input signal and the more negative regions. The amplifier is then said to be non-linear. This non-linearity is distortion, since the output waveform has a fundamentally different shape than the input waveform.
Make sense?
If you take a guitar and pluck a string, you'll get a note.
If you take the same guitar and touch the string lightly at the 12th fret (lightly!) and pluck the string hard, close to the bridge, and immediately remove the touching finger, you'll get a clear, bell like tone, at exactly an octave above the original tone. An octave is another way of saying twice the frequency.
This tone was there when you played the first note, but it was concealed by the first note, which is much louder, but which you have suppressed (along with most of the other harmonics), by touching the string at the same instant that you plucked it. The lower, first note is called the fundamental, or more rarely the first harmonic.
The high tone is called the second harmonic. If you do the brief touching at the 7th fret you will get the 3rd harmonic, the 5th fret will get you the 4th harmonic and the 4th fret the 5rd harmonic. The harmonics occur at 2, 3, 4, 5 times the frequency of the fundamental and so on.
Harmonics are also called overtones. It is these harmonically related tones, that lead to the formation of chords and all the rest of what we call classical harmony.
Unfortunately the addition of unwanted harmonics can also lead to an unpleasant sound or dissonance.
If you display a sine wave on an oscilloscope you will see a familiar wavy line. This is a graphical representation of a voltage. The voltage goes up and down with monotonous regularity. In fact we call it monotonic. It's got just one tone in it.
If you change the shape of the sinewave (distort it), even just the tiniest bit, although the picture on the oscilloscope shows just one (or a few) cycles, the change is repeated identically on all the cycles for as long as the sinewave lasts.
So that little change occurs at least once for each cycle of the original sinewave. In fact, in order to create a regular small change (distortion) in a sinewave it's necessary to add to it a second (and maybe 3rd, 4th, 5th...) sinewave of twice, three times, four times the frequency (and so on).
Here's a sinewave with quite large amounts of third and fifth harmonics added to it. Or to put it another way, quite a lot of distortion.
Any conceivable shape of distorted sinewave can be produced by the addition of harmonics.
This is how distortion of a sinewave (a single note) causes other harmonically related notes, or overtones, to be added to the original note, and why they occur at multiples of the original frequency.
w
If you take the same guitar and touch the string lightly at the 12th fret (lightly!) and pluck the string hard, close to the bridge, and immediately remove the touching finger, you'll get a clear, bell like tone, at exactly an octave above the original tone. An octave is another way of saying twice the frequency.
This tone was there when you played the first note, but it was concealed by the first note, which is much louder, but which you have suppressed (along with most of the other harmonics), by touching the string at the same instant that you plucked it. The lower, first note is called the fundamental, or more rarely the first harmonic.
The high tone is called the second harmonic. If you do the brief touching at the 7th fret you will get the 3rd harmonic, the 5th fret will get you the 4th harmonic and the 4th fret the 5rd harmonic. The harmonics occur at 2, 3, 4, 5 times the frequency of the fundamental and so on.
Harmonics are also called overtones. It is these harmonically related tones, that lead to the formation of chords and all the rest of what we call classical harmony.
Unfortunately the addition of unwanted harmonics can also lead to an unpleasant sound or dissonance.
If you display a sine wave on an oscilloscope you will see a familiar wavy line. This is a graphical representation of a voltage. The voltage goes up and down with monotonous regularity. In fact we call it monotonic. It's got just one tone in it.
If you change the shape of the sinewave (distort it), even just the tiniest bit, although the picture on the oscilloscope shows just one (or a few) cycles, the change is repeated identically on all the cycles for as long as the sinewave lasts.
So that little change occurs at least once for each cycle of the original sinewave. In fact, in order to create a regular small change (distortion) in a sinewave it's necessary to add to it a second (and maybe 3rd, 4th, 5th...) sinewave of twice, three times, four times the frequency (and so on).
Here's a sinewave with quite large amounts of third and fifth harmonics added to it. Or to put it another way, quite a lot of distortion.
Any conceivable shape of distorted sinewave can be produced by the addition of harmonics.
This is how distortion of a sinewave (a single note) causes other harmonically related notes, or overtones, to be added to the original note, and why they occur at multiples of the original frequency.
w
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Wavebourn makes a good point I think. The Fourier Transform is a mathematical construct which we use to represent the real world but it is somewhat artificial in a sense and it is at least in part responsible for the confusion. An active element like a vacuum tube does not add a series of harmonics in the sense that it plucks these harmonics out of thin air and combines them with the original signal to get an output signal. It simply does not respond to input linearly. That is the change in plate current is not proportional to the change in the grid to cathode voltage for all possible plate currents.
So if the bottom of the output waveform (sine wave for this example) does not swing as far below the steady state current as the top swings above then the waveform appears "squashed" on the bottom. This is the same waveform that you would get if you summed the input sine wave with one at twice its frequency and indeed it sounds like another softer note is being played an octave higher.
Where it gets interesting is when we consider that actual sounds generally involve not only multiple sine waves at different (not always harmonically related) frequencies but also random noise and various discontinuities. All of these things acting together make a very complex signal that one would be rather hard pressed to describe as a wave at all. What the ear actually detects is a complex continuously changing variation in pressure. All the intermodulation effects etc get very hard to sort out in any meaningful way so we tend to look for models to simplify the whole thing but it is an abstraction.
Interestingly it seems off hand that the distortions that look ugly on a scope tend to sound ugly too where as the more subtle looking ones tend to sound that way too even if they measure as a higher THD (at least in my limited experience). It would be very interesting, I think, if someone smarter than I were to look into some method of analyzing distortion in the time domain (which is where we actually live after all) rather than the frequency domain. Perhaps we are losing something critical in the integration process.
So if the bottom of the output waveform (sine wave for this example) does not swing as far below the steady state current as the top swings above then the waveform appears "squashed" on the bottom. This is the same waveform that you would get if you summed the input sine wave with one at twice its frequency and indeed it sounds like another softer note is being played an octave higher.
Where it gets interesting is when we consider that actual sounds generally involve not only multiple sine waves at different (not always harmonically related) frequencies but also random noise and various discontinuities. All of these things acting together make a very complex signal that one would be rather hard pressed to describe as a wave at all. What the ear actually detects is a complex continuously changing variation in pressure. All the intermodulation effects etc get very hard to sort out in any meaningful way so we tend to look for models to simplify the whole thing but it is an abstraction.
Interestingly it seems off hand that the distortions that look ugly on a scope tend to sound ugly too where as the more subtle looking ones tend to sound that way too even if they measure as a higher THD (at least in my limited experience). It would be very interesting, I think, if someone smarter than I were to look into some method of analyzing distortion in the time domain (which is where we actually live after all) rather than the frequency domain. Perhaps we are losing something critical in the integration process.
Interestingly it seems off hand that the distortions that look ugly on a scope tend to sound ugly too where as the more subtle looking ones tend to sound that way too even if they measure as a higher THD (at least in my limited experience). It would be very interesting, I think, if someone smarter than I were to look into some method of analyzing distortion in the time domain (which is where we actually live after all) rather than the frequency domain. Perhaps we are losing something critical in the integration process.
Hint: function generator and 2-ray oscilloscope.
I don't think Fourier is responsible for confusion, but our misunderstanding of what he actually said can create confusion. As soon as a sound reaches a microphone or passes through an amplifier it loses any discontinuities it may have had. The Fourier Series requires it to be periodic (unlike music), but the Transform relaxes that requirement. I suppose we could use wavelet transforms instead but they get more complicated.
All down the years there has always been a tendency for engineers (practical people) to regard the maths as pure theory and just a model, not reality. The trouble is that in many cases the maths is actually expressing reality, which the engineers sometimes wanted to deny (e.g. the sideband issue of about a century ago, the telegraph equation - in both cases reality was as the maths said, but not what the engineers thought it could be).
All down the years there has always been a tendency for engineers (practical people) to regard the maths as pure theory and just a model, not reality. The trouble is that in many cases the maths is actually expressing reality, which the engineers sometimes wanted to deny (e.g. the sideband issue of about a century ago, the telegraph equation - in both cases reality was as the maths said, but not what the engineers thought it could be).
Then they can just measure the sound coming out of the speakers (in anechoic or outside).
You could argue that the audio waveform is merely a coding device for transferring the fundamental and harmonic/overtone components from the musical instruments to the cochlea. If the hairs are only sensitive to amplitude and frequency, not phase, then an infinite number of waveforms are equivalent in that they would sound the same. I believe timing/phase does become relevant at lower frequencies. Anyway, the simplest way of preserving the signal is to preserve the exact coding too i.e. the waveform.
My experiments (as a teenager in the 70s) with building a tone generator using an array of sequenced slide pots to construct a waveform told me that timbre doesn't correlate to waveshape at all, and there are many waves that are shaped very differently that do indeed sound the same. My conclusion is that the relative phase of partials in a complex tone conveys no information whatsoever.
Except for binaural location, maybe? And of timbre, when sounds have attack/decay phases?
Yes. An amp isn't supposed to generate harmonics, its supposed to amplify the signal.
It's entirely possible to create dissonances with an instrument. Simply play a few notes all separated by only a semitone simultaneously. In fact the beauty of an electric guitar is that it is possible to create a sustained, louder, more unpleasant noise with it with greater ease than by virtually any other means.
w
Yes,thats why i found very strange for me ( as a newbie ) reading about AMPs and harmonics. For me the harmonics come from oscilators and musical instruments,but as i can see from the posts in tihis thread ,the AMPs also produce them (unfortunatly ?) .
THANKS
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